Scheduling flexible flowshops with sequence -dependent setup times

Graduate Theses, Dissertations, and Problem Reports

2001

Scheduling flexible flowshops with sequence -dependent setup

Scheduling flexible flowshops with sequence -dependent setup

times

times

Kanchana Sethanan

West Virginia University

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Sethanan, Kanchana, "Scheduling flexible flowshops with sequence -dependent setup times" (2001). Graduate Theses, Dissertations, and Problem Reports. 2349.

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SCHEDULING FLEXIBLE FLOWSHOPS

WITH SEQUENCE DEPENDENT SETUP TIMES

Kanchana Sethanan

Dissertation submitted to the

College of Engineering and Mineral Resources

at West Virginia University

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in

Decision Sciences and Production System

Wafik H. Iskander, Ph.D., Chair

Alan R. McKendall, Jr., Ph.D.

John L. Harpell, Jr., D.B.A.

Majid Jaraiedi, Ph.D.

Ralph W. Plummer, Ph.D.

Department of Industrial and Management Systems Engineering

Morgantown, West Virginia

2001

Keywords: Flexible Flowshop, Hybrid Flowshop, Dependent Setup Times,

Tabu Search, Heuristics

ABSTRACT

Scheduling Flexible Flowshops with Sequence Dependent Setup Times

Kanchana Sethanan

This dissertation addresses the scheduling problem in a flexible flowshop with sequence-dependent setup times. The production line consists of S production stages, each of which may have more than one non-identical (uniform) machines. Prior to processing a job on a machine at the first stage, a setup time from idling is needed. Also sequence dependent setup times (SDST) are considered on each machine in each stage. The objective of this research is to minimize the makespan. A mathematical model was developed for small size problems and two heuristic algorithms (Flexible

Flowshop with Sequence Dependent Setup Times Heuristic (FFSDSTH) and Tabu

Search Heuristic (TSH)) were developed to solve larger, more practical problems. The FFSDSTH algorithm was developed to obtain a good initial solution which can then be improved by the TSH algorithm. The TSH algorithm uses the well-known Tabu Search metaheuristic. In order to evaluate the performance of the heuristics, two lower bounds (Forward and Backward) were developed. The machine waiting time, idle time, and total setup and processing times on machines at the last stage were used to calculate the lower bound. Computational experiments were performed with the application of the heuristic algorithms and the lower bound methods. Two quantities were measured: (1) the performance of the heuristic algorithms obtained by comparing solutions with the lower bounds and (2) the relative improvement realized with the application of the TSH algorithm to the results obtained with the FFSDSTH algorithm. The performance of the heuristics was evaluated using two measures: solution quality and computational time. Results obtained show that the heuristic algorithms are quite efficient. The relative improvement yielded by the TSH algorithm was between 2.95 and 11.85 percent.

iii

ACKNOWLEDGEMENTS

I am deeply grateful to my dissertation advisor, Dr. Wafik Iskander, who spent

numerous hours to share his knowledge and intelligence. He continuously provided

valuable guidance, comments, and encouragement throughout this work. He has

always been available for help and advice in a friendly atmosphere that inspired

creativity and motivation. Without his help, I would have never finished this research.

I am also grateful to my dissertation committee, Dr. John Harpell, Dr. Alan

McKendall, Dr. Majid Jaraiedi, and Dr. Ralph Plummer, for their positive comments and

suggestions, which greatly improved the quality of this research.

I am profoundly grateful to my dearest friends in Thailand who always gave me

excellent encouragement throughout my graduate studies in the USA. I also thank Thai

students and fellow graduate students in the Industrial and Management Systems

Engineering Department at West Virginia University, who made my life and stay in

Morgantown such a joyful and truly exceptional experience.

Special thanks to the department of Industrial and Management Systems

Engineering, West Virginia University, which furnished hospitality for learning and

conducting research and provided me with financial support throughout my graduate

years in the department.

Finally, my deep appreciation goes to my parents, sisters, and brother, no matter

how far you were, you were always there for me. Your endless love, confidence, great

support, and excellent encouragement were crucial to my accomplishments and my

iv

Dedicated to Luang Por Prarajchabhavanavisuthi, my parents Sunee and Pichai Sethanan, my sisters Wachiraporn and Amornrat Sethanan,

v

LIST OF TABLES... viii

LIST OF FIGURES... xi

CHAPTER 1: INTRODUCTION... 1

1.1 Background... 1

1.1.1 Scheduling ... 1

1.1.2 The Place of Scheduling within an Organization... 4

1.1.3 Classification of Sequencing Problems... 6

1.1.4 The General Flowshop Scheduling Problem... 7

1.1.5 A Flexible Flowshop Environment ... 9

1.1.6 Dependent Setup Times... 9

CHAPTER 2: STATEMENT OF THE PROBLEM... 13

2.1 Introduction ... 13

2.2 Manufacturing Background ... 14

2.3 Problem Statement ... 15

2.4 Assumptions ... 16

2.5 Research Objectives... 17

CHAPTER 3: LITERATURE REVIEW... 18

3.1 Introduction and Overview ... 18

vi

3.3 Flowshop Scheduling Models... 21

3.3.1 Flowshop Scheduling Models without SDST Considerations... 21

3.3.2 Flowshop Scheduling Models with SDST Considerations... 25

3.3.3 Applications of Tabu Search to the Flowshop Scheduling Problem ... 32

CHAPTER 4: EXACT ALGORITHM... 39

4.1 Introduction ... 39

4.2 Mathematical Formulation... 39

CHAPTER 5: HEURISTIC ALGORITHMS... 48

5.1 Phase 1: Obtaining an Initial Solution Using the FFSDSTH Algorithm... 48

5.1.1 Start Time Determination ... 54

5.1.2 A Detailed Description of the FFSDSTH Algorithm... 56

5.2 Illustration of the FFSDSTH Algorithm ... 74

5.3 Phase 2: Improving the Initial Solution Using the TSH Algorithm ... 98

5.3.1 Implementing the TS Heuristic with the FFs(Qm1, Qm2,…,Qms)/Sipm/Cmax Problem... 98

5.3.2 Tabu List ... 101

5.3.3 Neighborhood Size... 104

5.3.4 Tabu Restriction ... 105

5.3.5 Admissible Moves ... 110

CHAPTER 6: LOWER BOUNDS... 121

6.1 Introduction ... 121

6.2 Lower Bound Determination... 121

6.2.1 Forward Method ... 125

vii

6.3 Illustration of the Lower Bound Calculations... 132

CHAPTER 7: COMPUTATIONAL EXPERIMENTS... 143

7.1 Introduction ... 143

7.2 Comparison of the Results of Heuristic Algorithms with the Lower Bounds... 145

7.3 Comparison between the FFSDSTH Algorithm and the TSH Algorithm ... 154

CHAPTER 8: CONCLUSIONS AND RECOMMENDATIONS... 158

8.1 Introduction ... 158

8.2 Summary of the Research ... 158

8.3 Contribution of the Research ... 160

8.4 Recommendations of for Future Research... 160

REFERENCES... 162

APPENDICIES... 167

APPENDIX A ... 168

APPENDIX B ... 174

viii

LIST OF TABLES

Table 3.1 Summary of Previous Research on

FFS Scheduling Problems... 30

Table 4.1 The Notation Used in the Mixed Integer Programming Model... 41

Table 4.2 Speeds of Machines at Each Stage... 44

Table 4.3 Processing Time of Each Product at Each Stage on the Standard Machine ... 45

Table 4.4 Setup Time from Idling for Each Product in Stage 1 ... 45

Table 4.5 Changeover Times between Products of Each Stage... 46

Table 5.1 Speeds of Machines at Each Stage... 74

Table 5.2 Processing Time of Each Product at Each Stage on the Standard Machine ... 75

Table 5.3 Setup Time from Idling for Each Product in Stage 1 ... 75

Table 5.4 Changeover Times between Products of Each Stage... 76

Table 6.1 Processing Times on the Fastest Machine at Each Stage and Changeover Times of Each Product on Each Stage... 133

Table 6.2 The Summations of Setup Time from Idling of the First Stage and Cumulative Processing Times of Each Product on the Fastest Machine from Stages 1 through S-1 ... 134

Table 6.3 The Values of CT(i) and β(i) Used to Calculate the Backward Lower Bound ... 138

Table 7.1 Values of Parameters Used with the Different Data Types ... 144

Table 7.2 Computational Results for Set 1 Type A: Heuristic Algorithms vs. Lower Bound ... 146

ix

Table 7.3 Computational Results for Set 1 Type B:

Heuristic Algorithms vs. Lower Bound ... 147

Table 7.4 Computational Results for Set 1 Type C:

Heuristic Algorithms vs. Lower Bound ... 147

Table 7.5 Computational Results for Set 1 Type D:

Heuristic Algorithms vs. Lower Bound ... 148

Table 7.6 Computational Results for Set 1 Type E:

Heuristic Algorithms vs. Lower Bound ... 148

Table 7.7 Computational Results for Set 1 Type F:

Heuristic Algorithms vs. Lower Bound ... 145

Table 7.8 Computational Results for Set 2 Type A:

Heuristic Algorithms vs. Lower Bound ... 145

Table 7.9 Computational Results for Set 2 Type B:

Heuristic Algorithms vs. Lower Bound ... 150

Table 7.10 Computational Results for Set 2 Type C:

Heuristic Algorithms vs. Lower Bound ... 150

Table 7.11 Computational Results for Set 2 Type D:

Heuristic Algorithms vs. Lower Bound ... 151

Table 7.12 Computational Results for Set 2 Type E:

Heuristic Algorithms vs. Lower Bound ... 151

Table 7.13 Computational Results for Set 2 Type F:

Heuristic Algorithms vs. Lower Bound ... 152

Table 7.14 Average of Computational Results for Sets 1 and 2 for all Data Types

Heuristic Algorithms vs. Lower Bound ... 152

x

Table 7.16 Relative Improvement Results for the Different Data Types in Set 2:.... 155

xi

LIST OF FIGURES

Figure 1.1 Information Flow Diagram in a Manufacturing System

(Pinedo, 1995)... 5

Figure 1.2 A Classification of Sequencing Problems ... 7

Figure 1.3 A Schematic Representation of a Flexible Flowshop Environment ... 10

Figure 3.1 The General Tabu Search Technique ... 35

Figure 3.2 Selecting the Best Admissible Move... 36

Figure 5.1 A Process Flow of the FFSDSTH and TSH Algorithms... 49

Figure 5.2 Flowchart of the Look Ahead Rule... 69

Figure 5.3 The Assignment of all Families to the First-Stage Machines... 82

Figure 5.4 Sequences of Products on the Machines at Stage 1 ... 88

Figure 5.5 Final Sequences of Products on the Machines at Stage 1... 90

Figure 5.6 Product Sequences on Machines at Stage 2 ... 96

Figure 5.7 Sequences of Products on Machines at the Last Stage... 97

Figure 5.8 Tabu List of a Move (s,m1,x,m2,y)... 102

Figure 5.9 Tabu Restriction when Jobs are Moved within a Machine ... 106

Figure 5.10 Tabu Restriction when Jobs are Moved between Machines ... 108

Figure 5.11 Flow Process of Moving Families between (or within) Machines at the First Stage... 115

CHAPTER 1 INTRODUCTION

1.1. Background

1.1.1 Scheduling

Scheduling is defined as the determination of relative position of jobs with

respect to a processing machine, including the assignment of definite times at which

processing occurs (Nawaz et al., 1983). Another view of scheduling is defined as

the "allocation of limited resources to jobs over time to perform a number of tasks"

(Baker, 1974, p. 2). Examples of resources include machines, operators, facilities,

computers, and transporters.

The problem of scheduling n jobs on m machines is one of the classical

problems in flowshop manufacturing that have been studied by researchers for

many years. Additionally, scheduling plays an essential role in the entire

manufacturing system. Production scheduling problems exist frequently in

production environments whenever resources are required to perform a set of

operations on jobs, and also when each operation can be accomplished in more

than one way (Randhawa & Kuo, 1997). Normally, there are two categories of

constraints that are commonly found in scheduling problems. First, there are

restrictions on the capacity of available resources and, second, there are

technological limits on the order in which jobs can be performed. Resource

constraints generally refer to processor capacities and limitations. Technological

constraints include alternative routing and precedence relationships. Alternate

routing means that the product can be produced on more than one processor, while

precedence constraints mean that the processor cannot process a specific job if

machines to various jobs and determination of the order in which the jobs will be

performed in order to optimize some criteria while satisfying the shop constraints.

Generally, there are three issues concerned with scheduling jobs on a set of

machines (Cheng & Sin, 1990):

1. What machine should be allocated to which job?

2. How to sequence the jobs in order to obtain the best schedule and meet the

constraints?

3. How can the reasonableness of a schedule be rationalized?

Hence, the scheduler wishes to optimize some measures of effectiveness

(such as minimization of makespan, mean flow time, lateness, or inventory) which

may vary from one situation to another, and to satisfy the production constraints

(e.g. production requirements, resource capacities, or operation procedures).

There are three issues that need to be specified when defining a scheduling

problem. These three issues, as presented by Cutright (1990), are:

1. Length of planning horizon,

2. Nature of tasks that will be scheduled, and

3. Criteria used to determine the best schedule.

Planning Horizon

Planning (time) horizons are usually classified as long-term,

intermediate-term (or medium-intermediate-term), and short-range. Long-intermediate-term planning typically involves

capacity and strategic issues and is the responsibility of the top management.

Management formulates policy-related questions such as gross labor-hours,

machine-hours, floor space, customer policies, new product development, research

long-term planning horizon is at least five years. This research assumes that all

long-term decisions have been made.

Once the long-term planning is made, operation managers begin

intermediate-range planning in order to meet the objectives of the firm, subjected to

a set of constraints imposed by the long-range planning decisions. Intermediate

planning involves activities such as the determination of production plans, workforce

levels, and forecasting product demand. Typically, the time horizon of short range

planning is in months. It is also assumed in this research that all of these decisions

have been determined and that workforce levels are fixed.

Short-range planning is dependent on both long and intermediate-range

planning decisions. Operations managers make these plans in conjunction with

supervisors and foremen who desegregate the intermediate plan into weekly, daily,

or hourly schedules. Short-range planning uses the production plan and workforce

level from the intermediate planning stage to determine job scheduling through the

resources in order to meet the criteria. The time horizon of short-range planning is

usually in days.

Nature of the tasks in the shop-floor system

The nature of tasks (or jobs) to be scheduled involves the following issues

and questions:

1. Can a job be split in case there are more than one processors capable of

performing it?

2. Are there several processors that can perform the same job?, or

Scheduling Criteria

Scheduling criteria are always a function of completion time of the jobs and

may also be a function of the due date. Examples include minimization of flow time,

lateness, or tardiness.

1.1.2 The Place of Scheduling within an Organization

The scheduling function must interface with many other important functions

in the manufacturing systems (e.g. production planning, master production planning,

material and capacity planning, etc.) as shown in the information flow diagram in

Figure 1.1. In order to provide the departments in an organization access to the

necessary scheduling information and enable the departments to provide the

scheduling system with relevant information (e.g. changes in jobs’ data and status

of machines), a management information system (MIS) or a decision support

system (e.g. forecasting, aggregate planning, and master production scheduling) is

probably needed (Chen, 1997). The process of scheduling begins with capacity

planning (also called long-term planning) which involves facility and equipment

acquisition. Intermediate planning includes aggregate and master production

planning. In the aggregate planning stage, decisions regarding the use of facilities,

people, and inventories are made. The master schedule then desegregates the

aggregate planning and develops an overall schedule for outputs. Short-term

schedules then translate capacity decisions, intermediate planning, and master

schedules into job sequences, specific assignments of personnel, machinery, and

ORDERS, DEMAND FORECASTS CAPACITY STATUS

SCHEDULING CONSTRAINTS MATERIAL REQUIREMENT

SCHEDULING

SCHEDULE PERFORMANCE

SHOP STATUS

DATA COLLECTION JOB LOADING

PRODUCTION PLANNING , MASTER SCHEDULING MRP, CAPACITY PLANNING SCHEDULING AND RESCHEDULING DISPATCHING SHOPFLOOR MANAGEMENT SHOPFLOOR

1.1.3 Classification of Sequencing Problems

To classify the major scheduling models, it is necessary to characterize the

configuration of resources and the nature of tasks. For instance, a model may

contain one resource type (single-stage problems) or several resource types

(multistage problems). If the set of tasks available for scheduling does not change

over the time, the system is called static. Conversely, if new tasks arise over time,

the system is called dynamic (Baker, 1974).

Day and Hottenstein (1970) depict a schema for classifying sequencing

problem as presented in Figure 1. 2. The framework shows that the sequencing

problems have been categorized according to the following components:

1. the nature of job arrivals, such as fixed batch size or continuous arrivals which

are given by a probability density function.

2. the number of machines involved, for instance, single machine production

(m = 1) or multi-machine production (m > 1), and

3. the nature of job route.

Further classification could be added to this figure which would include

characteristics such as setup time (e.g. dependent or independent of job sequence

on a given machine) and due date considerations.

This research focuses on a static scheduling problem: A flexible (hybrid)

flowshop with dependent setup times, which minimizes the maximum completion

time of all jobs. The jobs are available at time zero and have sequence dependent

setup times on machines at each production stage. All parameters such as

1.1.4 The General Flowshop Scheduling Problem

Flowshop scheduling problems can be classified into two categories: general

flowshop and permutation flowshop (Pinedo, 1995; Chen,1997). For the

permutation flowshop, each of the n jobs is processed on the machines (m =1, 2,

..., M) in the same order (Osman & Potts, 1989). On the other hand, the processing

sequences of jobs on machines from one stage to another could be different in the

general flowshop. In addition, flowshop scheduling may be classified as static or

SEQUENCING PROBLEM

MXN PROBLEMS

(FIX BATCH SIZE)

CONTINUOUS ARRIVALS (STOCHASTIC PROCESS) ONE MACHINE PROBLEM MULTI-MACHINE PROBLEM MULTI-MACHINE QUEUEING PROBLEM SINGLE CHANNEL QUEUEING PROBLEM PARALLEL ROUTING SERIAL ROUTING

(FLOWSHOP&JOBSHOP)

HYBRID SHOP PARALLEL ROUTING

SERIAL ROUTING

(FLOWSHOP&JOBSHOP)

HYBRID SHOP

(STATIC CASE)

(DYNAMIC CASE)

ROUTINGS

ROUTINGS

dynamic. In general, a static scheduling problem specifies a number of n jobs and

an optimal schedule is to be found with respect to the n jobs only (Dudek et al.,

1992), while a dynamic scheduling problem specifies that jobs are constantly

entering and leaving the job file according to some probability distribution in the

stochastic process (Day & Hotenstein, 1970).

The majority of the research published has thus far been devoted to the

static problem. The early work started with Johnson (1954) for the two-machine

case. Johnson's algorithm finds an optimal sequence that minimizes the maximum

flow time (called makespan) for all jobs. The simplicity of Johnson's method

encouraged other researchers to extend his idea in order to find optimal sequences

for the M-machine problem. For the M machine case, the Campbell, Dudek, and

Smith’s (1970) heuristic (CDS), which extends Johnson's algorithm, is considered to

be a very effective and robust heuristic (Ho & Chang, 1991). Generally, the static

flowshop problems have the following characteristics (Baker, 1974; Gupta, 1977;

Stafford and Tseng, 1990; and Sarin& Lefoka, 1993, and Pinedo, 1995).

1. Each machine can process at most one job at a time.

2. Each job can be processed on at most one machine at a time.

3. Preemption and splitting of any particular job are not allowed.

4. Jobs are processed on each machine in the same order.

5. All N jobs are available for processing at time zero.

6. All machines are available at time zero and are independent.

7. The processing time of each job on each machine is a known value.

1.1.5 A Flexible Flowshop Environment (FFS)

A flexible flowshop (FFS) is a generalization of the flowshop and the parallel

processor environments. A flexible flowshop is alternatively called a hybrid

flowshop or multiprocessor flowshop. In the most general setting of a flexible

flowshop environment, there are multiple stages (S stages), each of which consists

of m(s) (s = 1, 2, 3,…,S) parallel processors). A schematic representation of a

flexible flowshop environment is given in Figure 1.3. The processors in each stage

may be identical, uniform, or unrelated. Machines are uniform if the time to process

a job on any machine is a constant ratio of its processing time on other machines. In

other words, uniform machines are identical processors that do not have equal

speeds. Unrelated machines are machines for which the time to process a job on

any machine has no particular relationship of its processing time on any other

machine (Cheng & Sin, 1990). In a FFS environment, each job is processed first at

stage 1, then at stage 2, and so on. Normally, a job requires only one machine at

each stage and any machine can process any job.

1.1.6 Dependent Setup Times

Setup time is the time used to prepare the process of jobs on machines

(Allahverdi et al., 1999). Consequently, the requirements of setup times of jobs are

very common in many real manufacturing situations. This includes setting up tools

such as jigs and fixtures, cleanup, inspecting material, and positioning the jobs.

The issue of setup time has been of much interest in the past few decades.

According to the Goldratt Theory Of Constraint (TOC) (Goldratt, 1990), setup

reduction efforts can improve performance, but only if concentrated on production

bottlenecks or constraints. The total time for a machine can be classified as either

waste time (i.e., time spent processing material that cannot be converted into

throughput; for instance, time to process products for which there is no demand). It

is possible to improve the efficiency or capacity of a resource by reducing idle time

and waste time, cutting or reducing the total setup time, and reducing the production

time per unit of the product.

STAGE 1 STAGE 2 STAGE S

………... ………. ………..

:

:

:

:

:

:

:

:

:

Typically, there are two categories of setup times. In the first category,

setup time is sequence independent. That is, i.e., it depends only on the job to be

processed. In the second, setup time is sequence dependent as it depends on both

the job to be processed and the preceding job. Another view of setup time

classification adopted by Randhawa and Kuo (1997) includes: (1) processor

dependent, (2) product dependent, and (3) both processor and product dependent.

Processor dependent setup time deals with the setup time that depends only on the

processor, regardless of the job type, while product dependent setup time refers to

the setup time that depends only on the product, regardless of the machine type.

Sequence dependent properties (e.g. setup times or costs) are considered

to be important factors in the manufacturing environment, especially, when a shop

floor is operated at or near its full capacity (Wilbrecht & Prescott, 1969). Sequence

dependent setups are commonly found both in a single machine type or a multiple

machine type. Even though there exists an enormous amount of research on the

flowshop scheduling problem, research study has rarely been conducted in the case

where setup times are sequence dependent (Simon Jr., 1992; Allahverdi, 1999).

Hence, the results of these research studies lack a practical solution for applications

that require the treatment of setup times. For this reason, dependent setup times

cannot be neglected and hence are considered in this research.

Sequence dependent setups occur especially in process industry

operations, where machine setup time is significant and is needed when products

change. The magnitude of setup time depends on the similarity in technological

processing requirements (routing and precedence relationships) for the successive

jobs (Srikan & Ghosh, 1986). Normally, similar technological requirements for two

current products processed on the machine are from the same family that consists

of a set of similar products (or jobs) in terms of processing, then the changeover

time between those two products is small. The changeover times depend on the

family of products. This type of production system can be found in many industries

such as pharmaceutical, cosmetic, chemical, and food and brewing industries. The

following are real life examples of dependent setup times:

1. In printing industry, the cleaning and setting of presses are dependent on the

color of ink and size of paper.

2. In textile industry, weaving and dyeing setup operations depend on jobs.

3. In brewing and food industry (for container and bottling section), settings are

changed when the containers or bottle sizes change.

This research focuses on the scheduling problem in a flexible flowshop with

sequence dependent setup times. A complete description of the problem is given in

CHAPTER 2

STATEMENT OF THE PROBLEM

2.1 Introduction

Nowadays, manufacturing companies are faced with market demands for a

variety of high quality products. These companies must, therefore, make their

production systems more flexible, reduce costs related to production, and respond

rapidly to demand fluctuations. Hence, companies need to have advanced techniques

and an increasingly high degree of automation.

Production and operation management has been an interesting topic in

manufacturing, especially in such areas as job scheduling and system control. The

development of production schedules is a remarkably important task in industry. Many

scheduling researchers have focused their research on sequencing and timing the

scheduling of multiple non-identical jobs through one or more machine stations (Egbelu,

1991). A challenge facing many manufacturing and service industries is job assignment

to parallel processors (e.g., workers or machines). Parallel processing is the situation

where a job can be processed by more than one processor, but only one processor can

actually work on one job. This type of production system where multiple products are

processed on parallel, non-identical machines is common in both manufacturing and

service industries. For instance, airline companies may assign one of several types of

airplanes to service a route. In industries such as semiconductor manufacturing, it is

common to find newer or more modern machines running side by side with older and

less efficient machines. Even though the older machines are less efficient, they may be

kept in the production lines because of their high replacement costs. The older

machines may perform the same operations as the newer ones, but would generally

plants assigning jobs to looms and paper plants assigning products to different paper

machines (Randhawa & Smith, 1995). So, even though those resources may be of

similar type, their production rates may be different. This research will focus on

scheduling non-identical jobs in a flexible flowshop (or hybrid) environment with

sequence dependent setup times as described in the following section.

2.2 Manufacturing Background

Nature of the Tasks in the Shop-floor System

In this research, production is restricted by resource and technological

constraints. Processors (or machines) can process the same jobs but differ in their

speeds. Thus, the production rate for the same job may be different between machines

at the same stage, which results in different production costs per unit of the product.

This research deals with the general flexible flowshop, with S production stages,

in which the job sequence may not be the same on each machine at each stage. The

problem on hand has several distinct product families, and within each family there are

different product types. Each production stage may be composed of more than one

machine. If a stage has multiple machines, all machines would be similar in function but

different in their performance. All products may be processed on any of the machines in

a stage. It is assumed that the slowest machine in each stage has the lowest production

performance for all products. The problem hence will be developed and solved for the

parallel processing case with uniform processors.

Each product i of family j requires PTime(j,i,s,m) units of processing time on

machine m of stage s. A production line requires a setup time to change over from one

product to another. Machine changeover is needed when the product changes both

within a family and between families. In this research, two types of machine

the changeover time required if the previous product belongs to the same family. On the

other hand, if the previous product was of a different family, a major changeover time

would be required. The changeover time for machine m of stage s between product i of

family j and product p of family q is denoted by ch(j,i,q,p,s). If j = q, then this

changeover time is minor, but if j ≠ q, then it is a major one. The changeover time in this research is assumed to be asymmetric. This means that ch(j,i,q,p,s) may not be

necessarily equal to ch(q,p,j,i,s). It is also assumed that changeover times are equal for

all machines in the same stage of a production line when changing from one product to

another, but the changeover time may be different between stages.

The processing on all stages is not preemptive, which means that a new product

cannot enter into the stage until the previous product has been completely processed.

2.3 Problem Statement

This research addresses the problem of scheduling jobs in a flexible flowshop in

which machines are uniform. A job used in this study is synonymous with an order and

represents an individual, distinct demand for a product. Each production stage may be

composed of more than one machine. Prior to processing a job on a machine in a

production line, there is an associated setup time. Setup times are considered

significant and typically depend on the sequence of the jobs through the processors.

The problem considered in this study is complex in three ways:

1. Even though the flexible flowshop scheduling problems have been studied by

several previous researchers, very few of them have considered both

products and families in their models. This research addresses products

which are grouped into families to be processed in a flexible flowshop

environment. There are different products within each family, and there are

2. Both major and minor setup times are considered. A major setup time is

required if a machine at any stage switches from one family to another. On

the other hand, a minor setup time is needed if the previous product belongs

to the same family.

3. The system consists of S stages of production. Each production stage may

consist of more than one non-identical (uniform) machines. The production

line may have different number of machines in each stage. The system can

produce a number of products and families, and all products and families can

be produced on every processor.

This research addresses the problem of scheduling all products on the machines

at the different stages in order to minimize the makespan.

2.4 Assumptions

The assumptions made in formulating the problem are as follows:

1. It is assumed that the decisions about production plans, workforce levels, and layout

of the facility have been made from the long and intermediate-range planning.

2. Production is make-for-stock; hence, there are no due dates associated with batches

or products.

3. All jobs and machines are available at the beginning of the scheduling process (at

time zero).

4. There are many stages in the flowshop production line. Each stage may have

several non-identical but uniform machines.

5. Jobs may not be necessarily scheduled in the same order in all stages.

6. Jobs can wait between two production stages (or stations) and the intermediate

7. Within the same product family, minor changeover times may not be equal between

products. Likewise, major setup times may not be equal between families.

8. Setup times for jobs on each machine are dependent on the order in which jobs are

processed, but it is also assumed that setup times are equal for all machines in the

same stage when changing from one product to another.

9. No job splitting is allowed. A job must be completely finished on one machine before

it can be manufactured on the succeeding machine.

10. There is no job preemption.

2.5 Research Objectives

The major objectives of this research are:

1. To formulate a mathematical model to solve the problem and to produce an optimal

schedule in order to minimize the total makespan.

2. To develop efficient scheduling heuristics to find approximate solutions for large-size

problems.

3. To evaluate the heuristics developed by comparing their results to good lower

CHAPTER 3 LITERATURE REVIEW

3.1 Introduction and Overview

This research focuses on a static sequencing of a flexible flowshop (FFS)

environment. In a FFS environment, there are S production stages with one or more

machines at each stage. Sequence dependent setup times (SDST) are considered on

each machine. A review of previous work on flowshop scheduling is performed, along

with a review of the SDST flowshop literature. Also, a review of the literature on the

application of the Tabu search (TS) algorithm relevant to this study is presented.

A popular notation used in scheduling problems has the form of α/β/γ. The first parameter (α) describes the machine environment and contains a single entry. The second parameter (β) is a field providing the details of processing characteristics and constraints. The β field may contain no entry, a single entry, or multiple entries. The last parameter (γ) contains the objective to be minimized and usually contains a single entry. Flowshop problems deal with m stages in series and with one machine in each stage,

and are denoted, in general, as Fm//Cmax when makespan is to be minimized. If there

are several processors in each stage and all of them are identical, the problem becomes

a flexible flowshop, denoted as FFs(Pm1,Pm2,…,PmS)//Cmax. If the machines are

uniform in the flexible flowshop, then Pms are replaced with Qms for s =1,2,…,S. When

setup times are involved, the notation becomes FFs(Pm1,Pm2,…,PmS)/sip/Cmax and

Fm/sip/Cmax for the flexible flowshop and regular flowshop problems, respectively. In

addition, if the setup time between job i and p depends on the machine, then the

subscript m is added, that is, it becomes sipm. A complete list of the notation used in this

Before reviewing the literature on flowshop scheduling, a review of the

methodology for solving sequencing problems in general is presented in the following

section.

3.2 Solution Methodologies for Scheduling Problems

After determining the context in which scheduling is being defined, the

methodology for selecting a "good" schedule solution is determined. Day and

Hottenstein (1970) state that there are four common approaches used to solve the static

scheduling problem. These approaches are described below:

3.2.1 Combinatorial approach

Combinatorial approaches are based on the changing of one permutation

to another by switching jobs around in order to optimize a given objective

function.

3.2.2 Enumerative optimal methods

The most general techniques are mathematical formulations (including

linear programming, dynamic programming, integer programming, or mixed

integer programming), and branch and bound methods.

Scheduling problems are typically represented as an optimization

problem subject to a set of constraints. The problem takes the form of a

mathematical model that expresses the desired objective subject to the

constraints set forth in the problem. However, there are many difficulties in

formulating mathematical models. These difficulties include the complexity of the

interactions among many variables in a system, the difficulty in the attempt to

optimize the schedule from the system, and the difficulty in gaining an agreement

among these variables on what is essential for the good of the system (Cutright,

Typically, the mathematical model for the problem is either too difficult or

too time-consuming to solve in reasonable time. Since the development of a

mathematical model is a time-consuming task and requires a thorough

understanding of the system being represented, it is necessary to find solution

techniques that are easy to implement even though they may not always lead to

an optimal solution. These techniques include heuristic approaches and Monte

Carlo sampling which are described below.

3.2.3 Heuristic approach

Generally, difficulties arise in solving scheduling problems. Exact solution

procedures may not exist or may be too expensive to apply for large-sized

problems. One then has to use procedures that yield good (but not necessarily

optimal) solutions. These methods are termed heuristics. Heuristic approaches

can be divided into:

1. exact solution to a relaxed problem such as LP relaxation and

Lagrangian relaxation,

2. local search procedures including search techniques such as tabu

search (TS), genetic algorithm (GA), or simulated annealing (SA), and

3. ad hoc decision rules.

3.2.4 Monte Carlo sampling

Monte Carlo method is a technique for the solution of a model using

random (or pseudo random) numbers. For this approach, a scheduling problem

is solved by taking random samples of feasible solutions and using the best of

3.3 Flowshop Scheduling Models

In order to discuss relevant research in the area of flowshop scheduling, the

topics reviewed are divided into three categories: (1) models without SDST

consideration, (2) models which explicitly consider SDST, and (3) previous work

concerned with TS application to solve the flowshop scheduling problems.

3.3.1 Flowshop Scheduling Models without SDST Considerations 3.3.1.1 General Flowshop Scheduling (Fm/ /Cmax)

The flowshop scheduling problem with no setup times has been

researched extensively over the past five decades. Work on these problems was

pioneered by Johnson (1954), who presented a simple algorithm for solving the

F2//Cmax problems to optimality in a polynomial time. A wealth of research then

followed but will not be covered here as it is not relevant to the problem at hand.

3.3.1.2 Flexible Flowshop Scheduling (FFs/ /Cmax)

A flexible flowshop environment consists of S production stages, each of

which having m(s) parallel machines, s =1,2,…,S. The machines in each stage

may be identical, uniform, or unrelated. This section reviews previous work

performed in a flexible flowshop environment without SDST considerations.

3.3.1.2.1 Exact Approaches

Two-stage cases: FF2(Pm1,Pm2)//Cmax

Arthanary and Ramaswamy (1971) were the first to develop the FFS

problem (Soewandi, 1998). They proposed a branch and bound algorithm for the

two-stage FFS problem in which there are m identical machines in stage 1 but

only one machine in stage 2, FF2(Pm1, Pm2 =1)//Cmax. They could optimally

solve problems with up to 10 jobs with reasonable computational effort.

According to Gupta (1988), the two-stage flowshop problem in which

NP-complete. He proposed a heuristic to solve a special case when there is

only one machine in the second stage in order to minimize the makespan,

FF2(Pm1,Pm2=1)//Cmax. Computational experiments showed that the

effectiveness of the proposed heuristic increases as the problem-size increases.

Gupta and Tunc (1991) considered the FFs(Pm1=1,Pm2)//Cmax and

established approximate solution algorithms. They also developed a branch and

bound algorithm using the heuristic solution as an upper bound on makespan.

Their results showed that when the number of machines at stage 2 is equal to or

greater than the total number of jobs, the Longest Processing Time (LPT)

scheduling rule yields optimal solutions. For the case in which the total number

of jobs is greater than the number of machines in stage 2, they developed two

heuristics to minimize the makespan. Computational results indicated that the

effectiveness of the algorithms increases with the increase of the total number of

jobs. For the cases in which the deviations of the heuristic makespans were

relatively large from the lower bounds, an improved branch and bound algorithm

was developed. The maximum number of jobs reported in their work was only

eight jobs.

Multiple stage cases (FFs(Pm1,Pm2,…,PmS)//Cmax)

Brah and Hunsucker (1991) and Ragendran and Chaudhuri (1992)

developed branch and bound algorithms for the FFs(Pm1,Pm2,…,PmS)//Cmax.

Both studies can solve only small-sized problems. Portmann et al. (1998) also

studied the FFs(Pm1,Pm2,…,PmS)//Cmax problem. They improved the lower bound

of Brah’s and reduced the number of branches used in the search tree. They

also used a genetic algorithm (GA) approach to improve the search. Their

computational experiments indicated that optimal solutions using their branch

could solve problems with up to five stages (3, 3, 1, 2, and 2 machines in stages

1 through 5, respectively) and 15 jobs with an average deviation of 3% from the

results of the branch and bound algorithm.

Moursli (1995) also investigated on the FFs(Pm1,Pm2,…,PmS)//Cmax

problem. He derived three improvements from Brah’s algorithm and three new

lower bounds. His computational experiments showed that his algorithm could

solve problems with up to 20 jobs to optimality. Both number of nodes

investigated and running time were drastically reduced in his approach. Another

study was done by Vignier et al. (1996). They developed a branch and bound

approach to solve FFs(Pm1,Pm2,…,PmS)//Cmax and solve problems with up to 15

jobs.

3.3.1.2.2 Heuristic Approaches

Two stage cases (FF2(Pm1,Pm2)//Cmax)

Lee and Vairaktarakis (1994) developed five new lower bounds for the

FF2(Pm1,Pm2)//Cmax problem. They also proposed a heuristic to solve the

FF2(Pm1,Pm2,…,PmS)//Cmax problem. However, their results were not reported.

In 1996, Guinet et al. studied the scheduling for the FF2(Pm1,Pm2)//Cmax

problems. They developed a heuristic and three lower bounds. The

computational results showed that the average gap compared between the

heuristic solution and lower bounds are less than 0.73%. Another study was

done by Haouari and Hallah (1997). They developed a new lower bound and

used the Simulated Annealing (SA) and TS approaches to solve the problems.

According to the solutions of these problems, the TS based heuristic yielded an

optimal solution for 35 % of the cases and an average relative error of only

0.82%. In 1998, Soewandi developed a new procedure, which he termed

FF3(Pm1,Pm2,Pm3)//Cmax problems. He also considered the two-stage FFS with

uniform machines at each stage (FF2(Qm1,Qm2)//Cmax) and developed a solution

procedure adapted from Johnson's rule. Additionally, he proved that his heuristic

has a worst case performance Bound1 (w.c.p.b) for the FF2(Qm1,Qm2) problem

as 1+max{( 1₁)( ,1), 1 , 1 1 1

∑

∑

−

,

)

2

,

)(

1

(

v

v

m

} where vm is the speed of machine m,

and ms is the number of machines in stage s. Further, he developed two

heuristics for FF3(Pm1=1,Pm2,Pm3=1)//Cmax. Riane and Artiba (1997) and Riane

et al. (1998) studied FF3(Pm1,Pm2,Pm3)//Cmax problems, and developed two

heuristics to cope with realistic problems. The experimental results indicated that

their heuristics can solve problems with up to 130 jobs with a relative errors less

than 1% of the lower bound.

Multiple stage cases : FFs(Pm1,Pm2, …, PmS)//Cmax)

In 1994, Ding and Kittichartphayak developed three heuristics for

scheduling in FFs(Pm1,Pm2, …, PmS)//Cmax. The computational results showed

that one of their heuristics, called the combined approach, is the best and can

solve problem sets with number of jobs up to 8 with an average error less than

3% of the optimal solutions.

Multiple stage cases : FFs(Qm1,Qm2, …, QmS)//Cmax)

A multi-stage FFS scheduling problem in which jobs are identical and

machines are uniform at each stage was considered by Verma and Dessouky

(1999) with the objective of minimizing the makespan. They compared the Latest

Start Time (LST) rule with other heuristics: the Fastest Available Machine

1

An index that indicates the deviation of the performance values yielded by an algorithm, in the worst case, from the optimal solution for a given problem, or in some cases, from the values of the best known solutions or lower bounds.

Heuristic (FAMH), the Earliest Completion Time Heuristic (ECTH), and the Mix

Heuristic (MH). Their results indicated that the FAMH had a worst case absolute

bound that was twice as large as the ECTH, LSTH, and MH heuristics.

3.3.2 Flowshop Scheduling Models with SDST Consideration 3.3.2.1 General Flowshop Scheduling (Fm/ sipm /Cmax)

Allahverdi et al. (1999) presented a review of scheduling problems

involving setup considerations. They classified scheduling into batch and

non-batch, sequence-dependent, and sequence-independent setup. They also

summarized the results from the existing research and provided guidelines for

future research.

3.3.2.1.1Exact Approaches

Two-machine cases (F2/sipm /Cmax)

Prior to the research of the multiple machine problem, the two-machine

scheduling problem had been investigated by several researchers (e.g. Corwin &

Esogbue, 1974; Gupta, 1986, etc.). Corwin and Esogbue (1974) considered two

different flowshop scheduling problems with one of the machines having no setup

times. The objective of their study was to find the minimum makespan. After

establishing the optimality of permutation schedules, they solved the problem

using a dynamic programming formulation. Their findings showed that, from

computational standpoint, their formulation was comparable to that of the

traveling salesman problem (TSP). On the other hand, Gupta (1986) formulated

the Fm/sipm,no wait/Cmax problem as a TSP for the case in which jobs are

processed continuously through the shop. He showed that the flowshop

scheduling problem with SDST is NP-hard for the cases of limited or infinite

results from the TSP formulation of the continuous processing case were used to

describe an approximate solution for the cases in which the storage spaces were

limited or finite.

In addition to Corwin & Esogbue’s and Gupta’s studies, one of the studies

of Szwarc and Gupta (1987) was in terms of a special flowshop scheduling

problem with sequence dependent additive setup times. They developed a

polynomially bounded approximate method with the objective of minimizing

makespan.

Multiple machine cases (Fm/ sipm /Cmax)

Excellent efforts to solve the SDST for the m-machine flowshop problem

to optimality were performed by Srikar and Ghosh (1986). They developed a

method to reduce the number of constraints and binary variables in a MILP

formulation of the m-machine flowshop in order to minimize the makespan. They

could solve problems with up to six machines and six jobs; however, the time

required to solve problem was too large (22 minutes of CPU on a Prime 550

computer). Stafford and Tseng (1990) later discovered an error in Srikar and

Ghosh's model. They corrected it and solved the problem using LINDO. They

developed new MIP formulations for the regular flowshop problem and for the no

intermediate queues (NIQ) flowshop problem.

Exact optimization schemes are mostly based on the application of a

branch and bound (B&B) algorithm. The important part of a successful B&B

procedure lies in the computation of the lower bounds. In 1997, Rios-Mercado

developed several inequalities for two MIP formulations of the Fm/sipm/Cmax

problem. He used a branch and cut (B&C) procedure and found that this

main difference between the B&C and B&B procedures is that B&C algorithms

reduce the problem size (or a set of unevaluated nodes) by adjoining valid

inequalities (cutting planes or cuts). This, in turn, provides a stronger linear

programming-representation.

Recently, Rios-Mercado (1997) and Rios-Mercado and Bard (1999)

presented a branch and bound scheme for the SDST permutation flowshop

scheduling problem in order to minimize the makespan. Their algorithm included

the implementation of lower bounds and upper bounds and a dominance

elimination criterion, and yielded a significantly better performance over previous

work. They also could solve 100%, 43%, and 23% of 10-, 15-, and 20-job

problems, respectively, within a 1 % optimality gap. Gupta (1982) proposed a

branch and bound algorithm for the solution of the SDST flowshop with the

objective of minimizing the total setup times of machines. Unfortunately, the

computational results from the experiments were not reported. Because of the

complexity of the multiple machine scheduling problem, thus far no approach has

been found to solve the SDST flowshop to optimality for large-size problems.

3.3.2.1.2 Heuristic Approaches

Heuristic algorithms for the Fm/sipm/Cmax problem were developed by

Simons (1992), Rios-Mercado (1997), and Rios-Mercado (1999). Simons (1992)

developed four heuristics and compared them with three existing approaches (or

benchmark) that represent generally practiced approaches to scheduling in this

environment. However, only two of their proposed heuristics (called SETUP and

TOTAL) produced better results than the other heuristics tested. In addition,

computational experiments showed that problems with up to 15 machines and 15

Evidently, the most relevant work on heuristics for the Fm/sipm/Cmax

problem was conducted by Rios-Mercado (1997; 1999). They developed two

heuristics called HYBRID and GRASP to solve the problem. Experimental

results showed that the HYBRID heuristic outperforms GRASP when the number

of machines is small and when setup time fluctuations are large.

Moreover, Rios-Mercado and Bard (1998) made a comparison between

Simons's and Rios-Mercado and Bard's heuristics in relation to the Fm/sipm/Cmax

problems and concluded that, in general, Rios-Mercado and Bard’s heuristics

outperformed Simons’s SETUP heuristic. Nonetheless, in terms of better

solutions for the cases in which both setup and processing times are identically

distributed, Simons’s SETUP heuristic is relatively superior to Rios-Mercado and

Bard’s algorithms.

Another performance measure investigated by several researchers is the

minimization of the sum of weighted tardiness. Scheduling jobs on parallel

machines with SDST considerations were considered by Lee and Pinedo (1997).

They developed a three-phase heuristic, and a local search technique using SA

that is applied at the last phase. Additionally, Randhawa and Smith (1995)

investigated the factors that affected scheduling environments consisting of

parallel and non-identical processors. These factors are the processing capacity

relationships, sequencing and assignment rules, job sizes, and demand

distributions. They measured the effects of variables by comparing the mean flow

time, processor utilization spread, and proportion of tardy jobs. Computational

experiments showed that, setup times and system loading parameters were

3.3.2.2. Flexible Flowshop Scheduling (FFs/sipm /Cmax)

To date, no literature in the flexible flowshop with sequence dependent

setup time has been found. However, some literature is available on flexible

flowshops with independent time for the FFs(Pm1,Pm2,…,PmS)//Cmax problem as

presented below.

Setup times may simply be included in the processing times in the

situations where the entire batch of products is processed on one machine.

Conversely, if the same batch of products is partly assigned to several machines,

the same amount of setup time is still needed for the machines they are partly

assigned to and cannot be simply added to the processing times.

Li (1997) considered a two-stage FFS with a single machine at the first

stage and several identical machines at the second stage, and independent

setup times with the objective of minimizing the makespan, FF2(Pm1=1,Pm2)/ /Cmax.

He developed two heuristics adapted from previous work to solve the problem.

Gupta and Tunc (1994) developed polynomial heuristics for the two-stage FFS

scheduling problems in which there is only one machine in stage 1 and identical

machines in stage 2 but the number of machines at this stage is equal to or

larger than the total number of jobs. They also considered setup and removal

times independent from the processing times. The computational results

indicated that the effectiveness of the proposed algorithms increases when the

number of jobs increases. The contributions found in the literature for the FFS

scheduling problem are summarized in Table 3.1.

Exact algorithms based on branch and bound (B&B) and mixed integer

programming (MIP) were found in the literature to solve the problem. However,

the results of the computational experiments showed that B&B algorithms

because of their large size even for a small number of jobs and machines.

Hence, approximation methods such as TS have been paid attention to recently.

Table 3.1: Summary of Previous Research on FFS Scheduling Problems.

Problem Type References Methodology Problem size

FF2(Pm1,Pm2=1)//Cmax 1. Arthanary and Ramaswany (1971)

2. Gupta (1988)

Branch and Bound (B&B) Hueristic (w.c.p.b)

6-8 jobs 3 - (2 / m)

FF2((Pm1,Pm2=1)//Cmax Gupta and Tunc (1991) Heuristic

FF2(Pm1,Cm) //Cmax

(Cm= continuous flowshop)

Gupta (1997) Heuristic (w.c.p.b) 2- (1 / m)

FF2(Pm1,Pm2)//Cmax 1. Brah and Hunsucker (1991)

2. Lee and Vairaktarakis (1994) 3. Rajendran and Chaudhari (1992) 4. Moursli (1995)

5. Guinet et al. (1996) 6. Haouari and Hallah (1997) 7. Soewandi (1998) B&B Heuristic (w.c.p.b) B&B B&B Heuristic Heuristic Heuristic (w.c.p.b) ≤8 jobs 2- (1/max{m1,m2}) ≤ 8 jobs ≤ 20 jobs 2 –(1/max{m1,m2}) FF2(1,Pm2)//Cmax (independent setup is considered) Li (1997) Heuristic FF2(1,Pm2)//Cmax

(both independent setup and removal items are considered)

Table 3.1: Summary of Previous Research on FFS Scheduling Problems (continued).

Problem Type References Methodology Problem size

FFs(Pm1,Pm2,…,Pms)//Cmax 1. Lee and Vairaktarakis

(1994)

2. Moursli (1995) 3. Vignier et al. (1996) 4. Portmann et al. (1998) 5. Soewandi (1998)

6. Ding and Kittchartphayak (1996)

7. Novicki and Smutnicki (1996)

8. Franca et al. (1996)

9. Novicki and Smutnicki (1998)

Heuristic (w.c.p.b)

B&B B&B

B&B and B&B+GA Heuristic (w.c.p.b) Heuristic Heuristic (TS approach) Heuristic (TS approach) Heuristic (TS approach) S-(1/max{m1,m2}) - … - (1/max{m1-1,mS}

≤ 6 jobs for 5 stages

≤ 15 jobs

≤ 15 jobs for 5 stages 4- 1/max{m1,m2} – 1/m3 for Proc. SP1 10/3- 1/max{m1,m2} – 1/3m3 for Proc. SP2 8 jobs ≤ 500 jobs , 20 machines ≤ 50 jobs , 5 machines ≤ 150 jobs , 60 machines

FF2(Qm1,Qm2)//Cmax Soewandi (1998) Heuristic (w.c.p.b) {1+ (m-1)vm}/V

FFs(Qm1,Qm2,…,QmS)//Cmax

(for jobs are identical only)

3.3.3 Applications of Tabu Search (TS) to the Flowshop Scheduling Problem 3.3.3.1 Introduction and Overview

Tabu search is a heuristic designed for finding a near optimal solution for

combinatorial problems. It is considered as a metaheuristic (Hertez and Werra 1989,

1990, and Skorin-Kapov and Vakharia, 1993). This heuristic was first proposed by

Glover in 1989. It attempts to find a better solution than an initial. A key difference

between TS algorithm and other hill-climbing algorithms is that TS is not trapped at local

minima. The search process is provided with a mechanism that allows the objective

function to deteriorate and, in a controlled way, allows it to escape from local minima.

Researchers have shown that many combinatorial problems are NP-hard; hence,

near-optimal solutions are obtained. A heuristic method is often used to find an initial

solution which is then improved in an effort to find a near-optimal solution. Basically, the

application of TS is characterized by several components such as a move,

neighborhood, memory, initial solution, tabu list, aspiration level, and stopping criteria.

A move, a neighborhood, and a tabu list

A move is a function that transforms one solution to another. The subset of

moves applicable to a given solution generates a collection of solutions called the

neighborhood. TS begins with an initial solution which may be obtained from a heuristic

or from a random generation. At each step, the neighborhood of the current solution is

examined in order to find an appropriate neighbor. Typically, there are two fundamental

methods to examine an appropriate neighborhood. The first method is to examine the

entire neighborhood and select the best neighbor. This method is appropriate for

problems with small neighborhoods. The second method, which is useful with large

neighborhoods, is to examine a smaller neighborhood determined by some appropriate

technique. A trade-off exists between the effort spent in searching the neighborhood

performed and the resulting solution becomes the new current solution to initiate the

next iteration. The search allows for moves that yield solutions inferior to the best

solution obtained so far in order to avoid being trapped at a local optimum.

Since the search always chooses the best new movement, it may well fall back

into the local minimum from which it previously emerged. At any stage of the process, a

tabu list of mutation that the procedure is not allowed to perform is kept. The goal of

utilizing the tabu list is to exclude moves that would bring us back to the point where we

were at some previous iterations and keep us trapped in a local minimum. To avoid

cycling, the reverse of a movement that has been recently performed is forbidden (tabu)

and inserted on the top of tabu list. All other entries are pushed down one position and

the bottom entry is deleted. In other words, a tabu list is operated as a FIFO strategy.

The length of the tabu list is an important parameter. If the number of entries in the tabu

list is too small, cycling may occur. Conversely, if the number of entries is too large, the

computation time may increase significantly. The tabu list may be of several types such

as position of jobs or pairs of jobs that may not be interchanged (Tillard,1990).

Memory

Normally, there are three types of memories: short-term, intermediate, and

long-term memories. A fundamental component of the TS algorithm is a short-term

strategy called “simple TS” (Glover,1989; Glover, 1990; Werra & Hertz, 1989).

The fundamental memory structure in the simple TS algorithm is the so-called

tabu list. As mentioned earlier, each move in a tabu list is memorized after each

iteration. The best move is selected among the set of candidates which are not in the

tabu list. Normally, a short-term memory is a method that keeps limited track of a search

trajectory in order to guide the search out of a local optimum. The functions of

intermediate and long-term memories are employed within tabu search to achieve

solution space produces good solutions, then it is good to intensify the search in that

region (intensification). Conversely, instead of inducing the search to focus more

intensively on regions that contain good solutions previously found, the long-term

memory (diversification) guides the process to regions that markedly contrast with those

examined so far.

Aspiration level condition

An improvement can be realized in the TS is due to the fact that too many

solutions may be forbidden. An aspiration level is defined as the value of the best

schedule obtained so far. The aspiration level provides flexibility to choose good moves

by allowing the tabu status of a move to be overridden, after comparing the values of the

schedules, if it seems desirable to do so. Criteria for removing the tabu status will be

expressed by aspiration level condition.

Stopping criteria

Stopping criteria are rules to stop the search. Some stopping rules are defined

such as maximum number of iterations, maximum computation time, maximum CPU

time, or the maximum number of iterations have been performed without improving the

best solution obtained so far. Figures 3.1 and 3.2 describe the process of the tabu

search with short-term memory (Glover, 1990).

3.3.3.2 Review of TS Applications

During the last two decades, the Tabu Search (TS) technique has been

found to be a remarkably effective approach to solve combinatorial optimization

problems. Barnes and Laguna (1993) reviewed some of the research related to

TS applications in production scheduling and provided synthesis of the TS

methods that have been employed. Some suggestions for future research were

Terminate Globally or Transfer

A transfer initiates an long term memory components (intensification or diversification).

Generate An Initial Solution

It may be obtained from: ! an improvement heuristic ! a randomization.

Create a candidate list of moves (neighborhood)

! It is either not tabu or it is. If it is tabu, it can be overridden by the aspiration criteria.

! Each move would generate a new solution from the current solution.

Choose the best admissible move by evaluation each candidate move

! Select the best admissible move leading to the next solution

! record it as the new best solution if it improves on the previous best.

(Note: Detail is presented in Figure 3.2)

Stopping criteria

Stop the search if:

! a specified maximum number of iterations between two improvements of the objective function has reached

! a specified maximum number of iterations has reached, or

! the last best solution was found

Update Admissibility conditions

! Update Tabu restrictions, and ! Update aspiration level criteria

allowing the tabu status of a move to be overridden under appropriate circumstances

STOP CONTINUE

Figure 3.2: Selecting the Best Admissible Move Evaluate each candidate move

Does the move give the better solution than any other move found from the set of admissible candidates?

Check Tabu Status

Is the candidate is forbidden (tabu)?

YES

YES NO

Check Aspiration level

Does the move meet the aspiration level?

Move is admissible

The move is recorded as the best admissible candidate.

YES

Candidate List Check

Is there any probability of bettermove left, or should candidate list be extended?

NO

NO

Record and Upda